Best Known (70, 70+13, s)-Nets in Base 27
(70, 70+13, 1575250)-Net over F27 — Constructive and digital
Digital (70, 83, 1575250)-net over F27, using
- (u, u+v)-construction [i] based on
- digital (16, 22, 177150)-net over F27, using
- net defined by OOA [i] based on linear OOA(2722, 177150, F27, 6, 6) (dual of [(177150, 6), 1062878, 7]-NRT-code), using
- OA 3-folding and stacking [i] based on linear OA(2722, 531450, F27, 6) (dual of [531450, 531428, 7]-code), using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- linear OA(2721, 531441, F27, 6) (dual of [531441, 531420, 7]-code), using an extension Ce(5) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,5], and designed minimum distance d ≥ |I|+1 = 6 [i]
- linear OA(2713, 531441, F27, 4) (dual of [531441, 531428, 5]-code), using an extension Ce(3) of the primitive narrow-sense BCH-code C(I) with length 531440 = 274−1, defining interval I = [1,3], and designed minimum distance d ≥ |I|+1 = 4 [i]
- linear OA(271, 9, F27, 1) (dual of [9, 8, 2]-code), using
- discarding factors / shortening the dual code based on linear OA(271, s, F27, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
- construction X applied to Ce(5) ⊂ Ce(3) [i] based on
- OA 3-folding and stacking [i] based on linear OA(2722, 531450, F27, 6) (dual of [531450, 531428, 7]-code), using
- net defined by OOA [i] based on linear OOA(2722, 177150, F27, 6, 6) (dual of [(177150, 6), 1062878, 7]-NRT-code), using
- digital (48, 61, 1398100)-net over F27, using
- net defined by OOA [i] based on linear OOA(2761, 1398100, F27, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2761, 8388601, F27, 13) (dual of [8388601, 8388540, 14]-code), using
- discarding factors / shortening the dual code based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- the expurgated narrow-sense BCH-code C(I) with length 14348908 | 2710−1, defining interval I = [0,6], and minimum distance d ≥ |{−6,−5,…,6}|+1 = 14 (BCH-bound) [i]
- discarding factors / shortening the dual code based on linear OA(2761, large, F27, 13) (dual of [large, large−61, 14]-code), using
- OOA 6-folding and stacking with additional row [i] based on linear OA(2761, 8388601, F27, 13) (dual of [8388601, 8388540, 14]-code), using
- net defined by OOA [i] based on linear OOA(2761, 1398100, F27, 13, 13) (dual of [(1398100, 13), 18175239, 14]-NRT-code), using
- digital (16, 22, 177150)-net over F27, using
(70, 70+13, large)-Net over F27 — Digital
Digital (70, 83, large)-net over F27, using
- t-expansion [i] based on digital (68, 83, large)-net over F27, using
- 3 times m-reduction [i] based on digital (68, 86, large)-net over F27, using
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2786, large, F27, 18) (dual of [large, large−86, 19]-code), using
- the primitive expurgated narrow-sense BCH-code C(I) with length 14348906 = 275−1, defining interval I = [0,17], and designed minimum distance d ≥ |I|+1 = 19 [i]
- embedding of OOA with Gilbert–VarÅ¡amov bound [i] based on linear OA(2786, large, F27, 18) (dual of [large, large−86, 19]-code), using
- 3 times m-reduction [i] based on digital (68, 86, large)-net over F27, using
(70, 70+13, large)-Net in Base 27 — Upper bound on s
There is no (70, 83, large)-net in base 27, because
- 11 times m-reduction [i] would yield (70, 72, large)-net in base 27, but